ASYMPTOTIC FORMULAS FOR THE LYAPUNOV SPECTRUM OF FULLY-DEVELOPED SHELL-MODEL TURBULENCE

The scaling behavior of the Lyapunov spectrum of a chaotic shell model for three-dimensional turbulence is studied in detail. First, we char acterize the localization property of the Lyapunov vectors in wave-num ber space by using numerical results. By combining this localization p roperty with Kolmogorov's dimensional argument, we deduce explicitly t he asymptotic scaling law for the Lyapunov spectrum, which in turn is shown to agree well with the numerical results. This shell model is an example of high-dimensional chaotic systems for which an asymptotic s caling law is obtained for the Lyapunov spectrum. Implications of the present results for the Navier-Stokes turbulence are discussed. In par ticular, we conjecture that the distribution of Lyapunov exponents is not singular at null exponent.